Time: 9h00, Wednesday, February 4, 2015

Location: Room 4, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

Abstract: Let  $R=k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables   over field $k$. Let $\G$ be a simple graph with vertex set   $\{x_1,\ldots,x_n\}$ and edge set $E(\G)$. The squarefree monomial ideal  

$$I(\G) = (x_ix_j|x_ix_j\in E(\G))\subseteq R$$  

is called the {\it edge ideal} of $\G$. We say that  $\G$ is {\it Cohen-Macaulay} (resp. {\it Gorenstein})  over field  $k$  if  $R/I(\G)$ is   Cohen-Macaulay (resp.  Gorenstein) over field $k$. In this thesis, we classify all Cohen-Macaulay graphs of girth at least 5  and all triangle-free Gorenstein graphs. By this classification, we give  a purely combinatorial characterization for  Cohen-Macaulay property of the second power of edge ideals and of saturation of the second power of edge ideals.